[DreamingOfClouds]

Claim. Research package research/p104-consolidated-root-theorem (builds on #6868, the chain-nested root of #6833). Under a stated model and an explicit hypothesis ledger, chain(9|9) minimises D_root = f(8L+1) + sum[f(8|M|+1) - sum f(8r)] (f = x ln x, 1418.8458829 per port at L = 18) among shortest roots. Included: Dieudonne N >= n+1 with a direct rigidity proof (no Katz theorem; char != 2 automatic over Z[1/2][i]); star optimality for N = 2..1199; the block-partition S identity (all 384 partitions of 18, best -4.8%); the cost formula derived from the #6833 invoice to 1e-40 (exact kernel rescales D_root by -0.78%, optimiser unchanged); dense x and y (exact L <= 8, all L for single-host fields); roots with one extra unit (floor 1276.0 against break-even 1120.2, under R1-R4); D-free bases as a verifier convention with D-jobs priced. No kappa claim, no word.

Scope (open assumptions and items).

• Assumed: rank-one units; each D_k once and affine; [P_k, U_k] = 0 and x_k, y_k inside M_k in the site frame; liveness R4; D-free bases (verifier convention); the whole-job/compiler contract behind the cost formula; quoted constants (S = 760, native total).
• Twisted site-frame masks: not forced by the verifier's rules; closed at L = 2 (no twisted root below 90.854 against 87.7147; twisted sites with |S| = 2 do not exist, Groebner basis {1} over Q); open for L >= 4.
• Shared-host fields and pivot rows at L >= 10: proved only for single-host fields (all L) and exact up to L = 8.
• Roots with e >= 2 extra units, and e = 1 with R2/R3 dropped; the e = 1 floor needs R1-R4.
• J-staircase lemma (constant bases plus uncredited D-jobs): priced with no gain, not proved.
• Outside the model: non-coordinate masks, rank-2 units, non-gadget implementations, border and scratch-clearing escapes.

How to verify. python3.12 -I -B research/p104-consolidated-root-theorem/check.py (about 25 s, standard library only; prints PASS_P104_CONSOLIDATED_ROOT_THEOREM; --twisted adds two Groebner decisions, needs Singular). Registered check: python3.12 -B scripts/research_package.py --package research/p104-consolidated-root-theorem prints PASS_REGISTERED_RESEARCH_PACKAGE.

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